249 research outputs found
An efficient multi-scale Green's Functions Reaction Dynamics scheme
Molecular Dynamics - Green's Functions Reaction Dynamics (MD-GFRD) is a
multiscale simulation method for particle dynamics or particle-based
reaction-diffusion dynamics that is suited for systems involving low particle
densities. Particles in a low-density region are just diffusing and not
interacting. In this case one can avoid the costly integration of microscopic
equations of motion, such as molecular dynamics (MD), and instead turn to an
event-based scheme in which the times to the next particle interaction and the
new particle positions at that time can be sampled. At high (local)
concentrations, however, e.g. when particles are interacting in a nontrivial
way, particle positions must still be updated with small time steps of the
microscopic dynamical equations. The efficiency of a multi-scale simulation
that uses these two schemes largely depends on the coupling between them and
the decisions when to switch between the two scales. Here we present an
efficient scheme for multi-scale MD-GFRD simulations. It has been shown that
MD-GFRD schemes are more efficient than brute-force molecular dynamics
simulations up to a molar concentration of . In this paper, we
show that the choice of the propagation domains has a relevant impact on the
computational performance. Domains are constructed using a local optimization
of their sizes and a minimal domain size is proposed. The algorithm is shown to
be more efficient than brute-force Brownian dynamics simulations up to a molar
concentration of and is up to an order of magnitude more
efficient compared with previous MD-GFRD schemes
A variational approach to modeling slow processes in stochastic dynamical systems
The slow processes of metastable stochastic dynamical systems are difficult
to access by direct numerical simulation due the sampling problem. Here, we
suggest an approach for modeling the slow parts of Markov processes by
approximating the dominant eigenfunctions and eigenvalues of the propagator. To
this end, a variational principle is derived that is based on the maximization
of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be
estimated from statistical observables that can be obtained from short
distributed simulations starting from different parts of state space. The
approach forms a basis for the development of adaptive and efficient
computational algorithms for simulating and analyzing metastable Markov
processes while avoiding the sampling problem. Since any stochastic process
with finite memory can be transformed into a Markov process, the approach is
applicable to a wide range of processes relevant for modeling complex
real-world phenomena
Variational approach for learning Markov processes from time series data
Inference, prediction and control of complex dynamical systems from time
series is important in many areas, including financial markets, power grid
management, climate and weather modeling, or molecular dynamics. The analysis
of such highly nonlinear dynamical systems is facilitated by the fact that we
can often find a (generally nonlinear) transformation of the system coordinates
to features in which the dynamics can be excellently approximated by a linear
Markovian model. Moreover, the large number of system variables often change
collectively on large time- and length-scales, facilitating a low-dimensional
analysis in feature space. In this paper, we introduce a variational approach
for Markov processes (VAMP) that allows us to find optimal feature mappings and
optimal Markovian models of the dynamics from given time series data. The key
insight is that the best linear model can be obtained from the top singular
components of the Koopman operator. This leads to the definition of a family of
score functions called VAMP-r which can be calculated from data, and can be
employed to optimize a Markovian model. In addition, based on the relationship
between the variational scores and approximation errors of Koopman operators,
we propose a new VAMP-E score, which can be applied to cross-validation for
hyper-parameter optimization and model selection in VAMP. VAMP is valid for
both reversible and nonreversible processes and for stationary and
non-stationary processes or realizations
Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
Inspired by the success of deep learning techniques in the physical and
chemical sciences, we apply a modification of an autoencoder type deep neural
network to the task of dimension reduction of molecular dynamics data. We can
show that our time-lagged autoencoder reliably finds low-dimensional embeddings
for high-dimensional feature spaces which capture the slow dynamics of the
underlying stochastic processes - beyond the capabilities of linear dimension
reduction techniques
Kernel methods for detecting coherent structures in dynamical data
We illustrate relationships between classical kernel-based dimensionality
reduction techniques and eigendecompositions of empirical estimates of
reproducing kernel Hilbert space (RKHS) operators associated with dynamical
systems. In particular, we show that kernel canonical correlation analysis
(CCA) can be interpreted in terms of kernel transfer operators and that it can
be obtained by optimizing the variational approach for Markov processes (VAMP)
score. As a result, we show that coherent sets of particle trajectories can be
computed by kernel CCA. We demonstrate the efficiency of this approach with
several examples, namely the well-known Bickley jet, ocean drifter data, and a
molecular dynamics problem with a time-dependent potential. Finally, we propose
a straightforward generalization of dynamic mode decomposition (DMD) called
coherent mode decomposition (CMD). Our results provide a generic machine
learning approach to the computation of coherent sets with an objective score
that can be used for cross-validation and the comparison of different methods
A Software for Particle-Based Reaction-Diffusion Dynamics in Crowded Cellular Environments
We introduce the software package ReaDDy for simulation of detailed
spatiotemporal mechanisms of dynamical processes in the cell, based on
reaction-diffusion dynamics with particle resolution. In contrast to other
particle-based reaction kinetics programs, ReaDDy supports particle
interaction potentials. This permits effects such as space exclusion,
molecular crowding and aggregation to be modeled. The biomolecules simulated
can be represented as a sphere, or as a more complex geometry such as a domain
structure or polymer chain. ReaDDy bridges the gap between small-scale but
highly detailed molecular dynamics or Brownian dynamics simulations and large-
scale but little-detailed reaction kinetics simulations. ReaDDy has a modular
design that enables the exchange of the computing core by efficient platform-
specific implementations or dynamical models that are different from Brownian
dynamics
Reversible Interacting-Particle Reaction Dynamics
Interacting-Particle Reaction Dynamics (iPRD) simulates the spatiotemporal
evolution of particles that experience interaction forces and can react with
one another. The combination of interaction forces and reactions enable a wide
range of complex reactive systems in biology and chemistry, but give rise to
new questions such as how to evolve the dynamical equations in a
computationally efficient and statistically correct manner. Here we consider
reversible reactions such as A + B C with interacting particles and derive
expressions for the microscopic iPRD simulation parameters such that desired
values for the equilibrium constant and the dissociation rate are obtained in
the dilute limit. We then introduce a Monte-Carlo algorithm that ensures
detailed balance in the iPRD time-evolution (iPRD-DB). iPRD-DB guarantees the
correct thermodynamics at all concentrations and maintains the desired kinetics
in the dilute limit, where chemical rates are well-defined and kinetic
measurement experiments usually operate. We show that in dense particle
systems, the incorporation of detailed balance is essential to obtain
physically realistic solutions. iPRD-DB is implemented in ReaDDy 2
(https://readdy.github.io)
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